Two basic ideas come to mind. We want suspects not have their life destroyed if they are found not guilty. And after you have done your sentence, you should be free to have a normal life again.

For some people and some cases this is not possible. Sometimes there is a publicly known person involved, like the case of Keith Bakker which is in the Dutch news today. He has been a lot on TV in the past. Another case was the killing of Pim Fortuyn, a well-known politician, where everybody knows the name of the killer and he is now somewhat forced to live in another country. Mostly the news talk about Jos B. or similar semi-private namings. People closely involved know who it is about, but it doesn't need to be on TV or in the newspaper.

mpol already answered part of the design here about the need for privacy for the people involved; it is a balancing of priorities, both transparency in the judicial system and that privacy are important and this is a way of getting both.

About your other question though: one of the courts states in an answer to one of their frequently asked questions, https://www.rechtspraak.nl/Organisatie-en-contact/Organisati..., that a copy of a verdict will always be anonymised. No suggestions are made for when you want to apply for an exception to that rule and I suspect it is not possible at all. I am also struggling to come up with a valid reason for such a request; if it is indeed impossible, that might not be a problem.

How do they handle requests to view information about court cases currently in progress?

Example: I own and live in a condo apartment in a housing complex where the legal entities for my building and four other buildings are all members of one overall legal entity for the complex. Some of the other legal entities, and some of their administrators, are currently suing the overall legal entity with various claims. The management notified all owners that this was happening, as they're required by law to do, but didn't give the full complaint. (They did name the individuals and legal entities who filed suit, including the administrator roles of the individuals.) I'd like to read the full details.

Where I live, I can go to the public library or the courthouse, search for the involved parties in a database, and read the specifics. I don't have to prove that I am indirectly a partial owner of the entity being sued (though I am); this is available to the general public.

How would this work in the Netherlands if public access is anonymized?

I imagine the reason your laws only requires a notification that a court case is happening, without details, is because all the information is already public anyway and you can look up anything you need, so I expect that if Dutch law has similar provisions, it would require more details to be shared with you. However, this is just guessing, I have no experience with this, sorry.